b ipartite i ndex c oding arash saber tehrani alexandros g. dimakis michael j. neely department of...

BIPARTITE INDEX CODING

Arash Saber Tehrani

Alexandros G. Dimakis

Michael J. Neely

Department of Electrical Engineering University of Southern California (USC)

Outline

• Index Coding Problem– Introduction– Bipartite model

• Our Scheme: Partition Multicast– Formulation

• Partition Multicast is NP-hard– Connection to clique cover

Index Coding Problem

• Introduced in [Birk and Kol 98], and further developed in [Bar-Yossef, Birk, Jayram, and Kol 06 and 11].

• Broadcast station

• Set of m packets P ={x1, x2, … , xm} from a finite alphabet X

• Set of n users U ={u1, u2, … , un}

• Each user demands exactly one packet

• Each user i knows a subset of packets denoted by Nout(ui) as side info

• Objective: Minimize the amount of broadcast data so that all users decode their designated packets.

Bipartite model for IC

• The system can be represented by a bipartite graph

• A directed edge from packet xj to user ui indicates that user ui demands packet xj.

• A directed edge from user ui to packet xj indicates that user ui knows packet xj as side info.

Index Coding Problem

• A solution of the problem– A finite alphabet WX

– an encoding function E: Xm WX

– each user ui is able to decode its designated packet from the broadcast message w and its side information.

• Optimal solution is HARD to compute.

Our Scheme: Partition Multicast

• When each user knows at least d packets as side information– We call d “minimum out-degree” or “minimum

knowledge”

• Then there are at most m – d unknowns for each user. • With transmission of m - d independent equations in

the form a1x1 + a2x2 + … + amxm where ai's are taken from some finite field F, each user can decode the packet it demands as shown in Ho et al. (Given that |F| is large enough)

Our Scheme: Partition Multicast

• Induced subgraph by a subset of packets S

X1

X2

X3

X4

U1

U2

U3

U4

U5

X1

X2

U1

U2

U3

Our Scheme: Partition Multicast

• We are looking for a partition (valid packet decomposition)

X1

X2

X3

X4

U1

U2

U3

U4

U5

X1

X2

X3

X4

|{X1,X2}| = 2, d1 = 1 |{X3,X4}| = 2, d1 = 1

X1+X2 X3+X4

Our Scheme: Partition Multicast

• Partition Multicast:

Our Scheme: Partition Multicast

• The scheme is optimal for known cases such as – Cliques– trees– Directed cycles

• It has cycle cover schemes proposed by Chaudhry et al. and Neely et al. as a special case and outperforms them.

Partition Multicast is NP-hard

• Undirected case:

– We want to find a partition for which the sum of minimum knowledge is maximized

– We call this problem “sum-degree cover”

U1, X1

U2, X2

U3, X3 U4, X4

U5, X5

X1U1

U2 X2

X3

X4

X5

U3

U4

U5

Partition Multicast is NP-hard

• Sum-degree cover and clique cover are equivalent– Partitioning a clique is strictly suboptimal

• For any graph T(GS) ≥1.

• If GS is a clique, then T(GS) = 1, i.e., the minimum knowledge d = |S| - 1.

– We need to show that• Solution of sum-degree cover gives the solution of clique cover• Solution of the clique cover gives the solution of sum-degree

cover

SD cover Clique cover

• Let the solution of SD cover be GS1, … , GSK induced by subsets S1, S2, …, Sk.

• Clique cover is also a graph partition where each subgraph requires exactly one transmission, so

• Consider subgraph GS1 with minimum knowledge d1. The complement of GS1 has maximum degree |S1| - d1 - 1.

• As is well known, any graph of maximum degree d has a vertex coloring of size d + 1.

SD cover Clique cover

• The complement of GS1 has a vertex coloring with |S1| - d1 color.

• Thus, GS1 has a clique cover of size |S1| - d1.

• That is• Repeating the same procedure over all k

subgraphs, gives

• Jointly with the previous inequality we get

Partition Multicast is NP-hard

• Maps an undirected graph G to a bipartite graph.

• Solve the partition multicast.• Find the clique cover of all partitions through

coloring of complements of the subgraphs.• Find the clique cover.

Conclusion

• We introduced the bipartite graph model for the index coding problem

• We presented a new scheme “partition multicast” for index coding problem.

• We introduced the sum-degree cover problem.• We showed that finding the optimal partition

is NP-hard. • Future work: finding a ‘good’ partition

Thanks, Questions?

Partition Multicast

• Partition or Cover:

– Let x S∈ 1, x S∈ 2

– Delete x from S1 to get set S1’

– New minimum knowledge for GS1, namely, d1’.

– |S1’| =|S1|-1 and d1-1 ≤ d1’ ≤ d1.

GS1 GS2 GSk

T(GS1)=|S1|-d1 T(GS2)=|S2|-d2 T(GSk)=|Sk|-dk

Our Scheme: Partition Multicast

• Bipartite case (Painful stuff)– For set S P,⊆ define GS = (US,S,ES) to be the subgraph induced by S:

– A valid packet decomposition is set of k disjoint subgraphs such that

– It can be checked that for a valid packet decomposition

Index Coding Problem

• A solution of the problem– A finite alphabet WX

– an encoding function E: Xm WX

– each user ui is able to decode its designated packet from the broadcast message w and its side information.

• The minimum coding length of the solution per input symbol:

where the minimum is over all encoding functions E.• Optimal broadcast rate